Question

I . (30%) For the system shown in Fig. 3 with Ge K, Gr. 1/[s (s +2), H l + ks, determine the values of K and k such that the system has a damping ratio of 0.7 and undamped natural frequency of 5 rad/sec. Then obtain the settling time in the unit-s response. Controller PlantC Fig. 3 - _ I IFI

Answer 1

step 1:

1+ks

Step 2: As K and 1/s(s+2) are in cascade so their equivalent is = K/s(s+2),

1+kS

Step 3: Eliminate the summing point, therefore tranfer function is:-

C(s)/R(s) = (K/s(s+2)) / (1+(K/s(s+2))*(1+ks))

on simplifying we get,

C(s)/R(s) = K / (s^2 +2s +K + Kks) = K / (s^2 + (2+Kk)s + K)

Now the characteristic equation is:

s^2+(2+Kk)s+K=0

?n = K^(0.5)

5 = K^(0.5)

K= 25

and

£ = 0.7 (given)

Now,

2£?n = 2+Kk   

2*0.7*5 = 2+25*k

k= 0.2

Now,

Settling time,

tn = 4/£?n

    = 4/0.7*5

   = 1.143